L(s) = 1 | + 2-s − 1.70·3-s + 4-s − 3.24·5-s − 1.70·6-s − 7-s + 8-s − 0.0783·9-s − 3.24·10-s − 4.04·11-s − 1.70·12-s + 3.87·13-s − 14-s + 5.55·15-s + 16-s − 6.97·17-s − 0.0783·18-s − 3.24·20-s + 1.70·21-s − 4.04·22-s − 4.26·23-s − 1.70·24-s + 5.55·25-s + 3.87·26-s + 5.26·27-s − 28-s − 6.78·29-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.986·3-s + 0.5·4-s − 1.45·5-s − 0.697·6-s − 0.377·7-s + 0.353·8-s − 0.0261·9-s − 1.02·10-s − 1.22·11-s − 0.493·12-s + 1.07·13-s − 0.267·14-s + 1.43·15-s + 0.250·16-s − 1.69·17-s − 0.0184·18-s − 0.726·20-s + 0.372·21-s − 0.863·22-s − 0.888·23-s − 0.348·24-s + 1.11·25-s + 0.760·26-s + 1.01·27-s − 0.188·28-s − 1.26·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5054 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5054 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4184631950\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4184631950\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 19 | \( 1 \) |
good | 3 | \( 1 + 1.70T + 3T^{2} \) |
| 5 | \( 1 + 3.24T + 5T^{2} \) |
| 11 | \( 1 + 4.04T + 11T^{2} \) |
| 13 | \( 1 - 3.87T + 13T^{2} \) |
| 17 | \( 1 + 6.97T + 17T^{2} \) |
| 23 | \( 1 + 4.26T + 23T^{2} \) |
| 29 | \( 1 + 6.78T + 29T^{2} \) |
| 31 | \( 1 - 2.78T + 31T^{2} \) |
| 37 | \( 1 + 6.34T + 37T^{2} \) |
| 41 | \( 1 + 2.63T + 41T^{2} \) |
| 43 | \( 1 + 8.78T + 43T^{2} \) |
| 47 | \( 1 + 9.31T + 47T^{2} \) |
| 53 | \( 1 + 5.89T + 53T^{2} \) |
| 59 | \( 1 + 0.510T + 59T^{2} \) |
| 61 | \( 1 - 12.1T + 61T^{2} \) |
| 67 | \( 1 + 6.92T + 67T^{2} \) |
| 71 | \( 1 - 4.02T + 71T^{2} \) |
| 73 | \( 1 - 10.0T + 73T^{2} \) |
| 79 | \( 1 - 9.94T + 79T^{2} \) |
| 83 | \( 1 + 3.24T + 83T^{2} \) |
| 89 | \( 1 + 13.9T + 89T^{2} \) |
| 97 | \( 1 - 6.78T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.302102883936681255996703854218, −7.32078046509571278137881156521, −6.63597641916640450145939870363, −6.07407115875079445374395740232, −5.19817208557314650395323631496, −4.64284239723708763038632214223, −3.77914392703761566422114610403, −3.19738940232253914371764237745, −2.00581902351187474212639035349, −0.31242819526500941749571406703,
0.31242819526500941749571406703, 2.00581902351187474212639035349, 3.19738940232253914371764237745, 3.77914392703761566422114610403, 4.64284239723708763038632214223, 5.19817208557314650395323631496, 6.07407115875079445374395740232, 6.63597641916640450145939870363, 7.32078046509571278137881156521, 8.302102883936681255996703854218